A Superstring Theory in Four Curved Space-Time Dimensions
关于霍金的英文简介
关于霍金的英文简介斯蒂芬·威廉·霍金,英国剑桥大学著名物理学家，是现代最伟大的物理学家之一，也是20世纪享有国际盛誉的伟人之一，下面是小编为你整理的关于霍金的英文简介，希望对你有用!斯蒂芬·威廉·霍金学术思想Time MachineBritish famous astrophysicist Stephen William Hawking following the recognition of the existence of aliens, but also issued a stunning discussion: he claimed that with the human flying into the future of the time machine, theoretically feasible, the required conditions include space In the wormhole or speed close to the speed of light of the spacecraft. However, Hawking also warned not to take time to go back to history, because "only crazy scientists, will want to go back to the past 'upside down causal'.Physicist Hawking, in a documentary about the universe, notes that mankind can build a spacecraft close to the speed of light and be able to enter the future. Hawking even said that he was worried about others as a "weirdo", so he did not dare to speak, do not want to talk about the time machine things, until the documentary after the generous discussion.Time gapAs for the key point of time machine, Hawking stressed that the so-called "four degrees of space", scientists named "wormhole." Hawking stressed that "wormhole" in our surroundings, but small to the naked eye can not see, they exist in space and time cracks.He pointed out that the universe is not flat or solid, close to the observation will find all objects will appear small holes or wrinkles, which is the basic physical law, and apply to time. Time also has subtle cracks, wrinkles and voids, than the molecules, atoms also small space is named "quantum bubble", "wormhole" exists in them.return to the pastHawking pointed out that the theory of time tunnel or "wormhole" can not only carry humans to other planets, if the wormhole at both ends of the same location, and time rather than distance, then the spacecraft can fly into, after flying still Close to the earth, just into the so-called "distant past". Because in the 4-degree space, 10 minutes may be n hours. But Hawking warned, do not take time to go back to history.Fly to the futureSteven William Hawking said that if scientists can build a spacecraft close to the speed of light, then the spacecraftwill inevitably because it can not violate the speed of light is the maximum speed limit law, resulting in the cabin time slows down, then the flight is equal to a week 100 years on the ground, it is equivalent to fly into the future.Hawking lift artificial satellite, for example, refers to the satellite in orbit, due to the impact of the Earth's gravity is small, the satellite time is slightly faster than the ground time. As a result, Hawking envisioned a large speed universe ship, can be accelerated in 1 second to 97,000 km per hour, 6 years to accelerate to 99.99% of the speed of light, faster than the history of the fastest ship Apollo 10 times. The passengers on board the ship is disguised to the future, making a real time travel.Four degrees of spaceEven in space, everything has a length of time, in time roaming, means through the "4 degrees space."Hawking, for example, pointed out that driving straight line is equal to the "1 degree space" in the road, and turn left or right is equal to plus "2 degrees space", as in the meandering mountain road up and down, it is equivalent to enter the "3 degree space " Crossing the time tunnel is to enter the "4 degrees space."Alien theoryStephen Hawking in the United States to explore the channel April 25, 20xx broadcast documentary "follow Stephen Hawking into the universe," said that the possibility of alien existence is great, but humans should not take the initiative to find them, make every effort Avoid contact with them.Hawking believes that, in view of the alien may be looted the Earth's resources and then sped away, the human initiative to seek contact with them "some too risky." "If the aliens visited us, I thought the results could be similar to that of Christopher Columbus in the American continent, which was not a good thing for the local Indians.However, there are many experts on the "alien threat theory" expressed doubts. They from the alien's wisdom and technology, and even human self-alien psychological role to illustrate the alien people do not pose a threat to the earth. Chinese linguist and mathematician Zhou Haizhong said that if aliens can come to Earth, that their civilization is far more than our human; the higher the degree of civilization, the lower the possibility of violent tendencies. He believes that the threat of aliens is completely unnecessary, because as long as the wisdom of life, their rationality determines how they treatother intelligent life; aliens and the future of the earth is able to peaceful coexistence, friendly cooperation And common development. Coincidentally, American astronomer David Morrison also said that if a civilization can exist for hundreds of thousands of years, then it must be more advanced than human. This civilization will certainly solve a series of problems we face, so there is no need to invade the earth. He even believes that aliens are "peace messengers" and are friendly and cute, and said humorously, "If the aliens visit, I will treat them well." Nobel Prize in physics, the American astrophysicist George Mute eyes, "alien threat theory" all the worries are unfounded.Interstellar immigrationHawking in August 20xx to accept the US intellectual video sharing site BigThink interview, and then exposing remarks, said the Earth will be destroyed within 200 years, and human beings want to continue to survive only one way: immigration planets.Hawking said that if humans want to continue, they must immigrate to Mars or other planets, and the Earth will sooner or later die. Hawking said: "Human beings have entered an increasingly dangerous period, we have experienced a number ofevents related to life and death.As the human gene carried by the 'selfish, greedy' genetic code, humans are a little bit of plunder of the earth resources, Humans can not put all the eggs in a basket, so they can not put a bet on a planet.Academic betsHawking liked some scientific propositions, gambling with other scholars, and sometimes became a scientific scholar.1. Can you find the Higgs boson?Hawking made a bet with Professor Gordon of the University of Michigan: CERN would not find the Higgs boson.Higgs boson is the famous British physicist Higgs and other colleagues after a long period of painstaking research, waiting for 48 years, only to find the Higgs boson, also known as "God particles."2. Does the black hole exist?Set of energy to study the black hole of Hawking, had worried that the black hole may be just a theoretical concept, and the reality does not exist. He became nothing when he was free, and in 1975 he was gambling with another physicist, Kip Thorne, whether the black hole existed.The presence of naked singularityIn 1991, Hawking also asked to open the gambling, the Thornand his standing in the same front, the bet on the side of the physicist Pei Shiji (John Preskill). The proposition at that time was whether the singularity should be surrounded by a black hole, but whether there was a "naked singularities" surrounded by a black hole.Hawking and Thorne bet: the naked spot does not exist, then with the Berez set the gambling, who lost to the other side to send a used to "cover nudity" T-shirts, write the appropriate clothes to lose words. Hawking amended his theory in 1997, pointing out that naked singularity may exist.High dimensional spaceAccording to the M theory proposed in the 1990s (a kind of superstring theory), the universe is eleven dimensions, composed of the plane of vibration. In Einstein, the universe is only four-dimensional (three-dimensional space andone-dimensional time), modern physics is that there are seven-dimensional space we can not see.How does the scientist explain the difference between what we have known and what may exist but not perceived? They make an analogy: an ant travels on a piece of paper, Or left, forward or backward. It is high and low meaningless, that is to say, the third dimension of the space is there, not known by the ants.Likewise, our world is made up of four-dimensional data (three spatial dimensions, one time dimension), and we are not aware of all other dimensions.According to the physicist's view there should be seven dimensions. Despite having so many dimensions, these dimensions are invisible, and they themselves are rolled together, known as compressed dimensions. In order to clarify this view, let us to ants as an example to start our imagination. We can imagine, the ants in the above walking paper roll up, until rolled into a cylindrical. If the ants walk along the paper wall, and finally it will return to the starting point, which is an example of compressed dimension. If you can take along the famous Mobius, the above phenomenon will occur, of course, it is three-dimensional, but if you walk along it, will always return to the starting point. Mobius is compressed from the perspective of the dimension, according to the physics it has three dimensions, but who walking in the above, can only be recognized as a dimension. This is a bit like the people on the left: up or down, but never come to an end. If the ant is not walking along the curved wall of the paper tube, it will never return to the original starting point. This is an example of two dimensions (or the kind of dimension we are perceived),and it is impossible to return to the original starting point along it.斯蒂芬·威廉·霍金主要成就Stephen William Hawking's research laid the groundwork for today's understanding of the black hole and the origin of the universe, but according to himself, he said that he was in the animated "The Simpsons" and the science fiction episode "Star Trek: Next Generation" (Star Trek: The Next Generation) is also wonderful.Hawking emphasizes that the universe does not need a Creator or "God" in the "Great Design", "philosophy is dead", which means that mankind will be detached from ignorantself-slavery, denying that pure philosophy and religion can really explain Naturally, it also shows that the major religions are only the ancient spiritual world to explore the unknown, the pursuit of immortal system, rather than the objective truth. With the progress of the times, human civilization is also catch up, not far behind, which is why generations of people of insight to the existence of life and the meaning of the universe. To solve these propositions should have been the task of the philosopher, but unfortunately the highly developed science makes the philosophy can not keep up. Hawking in the "big design" of the opening that "philosophy isdead" is the meaning.Hawking hopes to solve the mystery of the birth of the universe, the 1970s, Hawking quantum mechanics applied to explain the phenomenon of black hole, in the next 30 years, with quantum mechanics to explain the universe has become more difficult. Hawking wanted to find a set of theories that could explain the universe as a whole to illustrate the birth of the 13.7 billion years of the universe until now, but it has not been concluded for years even if it is infinitely close. According to his theory of quantum mechanics, the birth of the universe is the big bang produced, which is a compressed infinitely small but with large gravity of the material (also can be understood as the density of infinite) explosion products. The theoretical category of quantum mechanics can not explain how this process is going to be done. Why is it so? Hawking says "that must have a theory that can describe small-scale gravity."The latest scientific breakthrough is Hawking's colleague, Michael Smith of London's Queen Mary's College (Michael. Green) involved in the construction of the superstring theory, referred to as "string theory", which states that all particles and natural forces are actually in shock In the universe likea small object, to solve the Hawking has always wanted to try to answer the gravity problem, this theory must be established in the universe must have 9,10 or even greater than 11 dimensions, and human beings in the three-dimensional world may only One of the real ones of the universe ...A large number of scientists around the world are doing experiments in space and earth to prove string theory and from experiments to support Hawking's black hole theory and quantum theory. January 24, 20xx, the famous British scientist Professor Stephen Hawking once again with its blackhole-related theory shocked the physics, in a recently published paper admitted that "black hole does not exist," but "gray hole" indeed exist. In a paper entitled "Information Preservation and Weather Forecasting For Black Holes", Hawking points out that "black holes do not exist" because they can not find the boundaries of black holes. In order to solve the "firewall" problem in the new theory set "black hole does not exist", it does not really do not exist. The black hole of the boundary, also known as the "horizon", the classic black hole theory that the black hole outside the material and radiation can enter the black hole through the horizon, and any material and radiation within the black hole can not wear out horizons.Hawking's latest "gray hole" theory that material and energy in the black hole trapped after a period of time, will be re-released into the universe. He admitted in his essay that his initial knowledge of the horizon was flawed, and that light could cross the horizon. When the light flies the black hole core, its movement is like a person running on a treadmill, slowly through the outward radiation and shrink. "The classical black hole theory argues that any matter and radiation can not escape the black hole, and quantum mechanics suggests that energy and information can escape from the black hole." Hawking also pointed out that the interpretation of this escape process requires a gravity And other basic forces of successful integration of the theory. In the past hundred years, no one in physics has tried to explain this process.For Hawking's "gray hole" theory, some scientists expressed approval, it was skeptical. Joseph Polchinski, a theoretical physicist at the Cuban Institute of Theoretical Physics, points out that according to Einstein's theory of gravity, the boundary of the black hole is present, but it differs from the rest of the universe Not obvious. In fact, as early as 20xx, Hawking had made a similar statement. On July 21 of that year, Hawking pointed out at the 17th InternationalSymposium on General Theory of Relativity and Gravitation that the Black Hole was not "completely swallowed" around it, as he and most other physicists had previously thought, Some of the information that is sucked into the depths of the black hole may be released at some point.In 1973, Hawking said he calculated by the conclusion that the black hole in the formation of the process of its quality reduction, but also continue to be in the form of energy to the outside world radiation. This is the famous Hawking radiation theory, the theory mentioned in the black hole radiation does not include the black hole inside the material of any information, once the black hole is concentrated and evaporated disappear, all of which information will disappear, which is the so-called The "black hole paradox". This theory and quantum mechanics of the relevant theories appear contradictory. Because modern quantum physics finds that this material information is never completely gone.For more than 30 years, Hawking tried to explain this contradictory view with various speculations. Hawking has said that the quantum movement of the black hole is a special case, because the gravity in the black hole is very strong, quantum mechanics at this time is no longer applicable. Hawking'sargument does not convince the scientific community of skeptical scholars. It now appears that Hawking finally gave this year's contradictory view of a more convincing answer. Hawking said the black hole never completely shut itself - Hawking radiation, they in a long period of time gradually to the outside world to radiate more and more heat, then the black hole will eventually open themselves and release the material contained in the information.On August 16, 1616, Jeff Steinhauer, a professor at the Israel Institute of Technology in Haifa, proved the quantum effect of Hawking radiation in a paper published in the journal Nature Physics. He made a sound black hole instead of a light black hole, using a long tube with sound particles, the phonon "horizon". In 20xx, Professor Steinhall found that the phonemes were randomly generated in the horizon. In his latest results, Steinhouse proved that these phonons were one of a pair of related phonons, thus proving the quantum effect of Hawking radiation.---来源网络整理，仅供参考。
More Superstrings from Supergravity
a rXiv:h ep-th/98547v216J un1998hep-th/9805047More Superstrings from Supergravity Clifford V.Johnson ♭Department of Physics University of California Santa Barbara CA 93106,U.S.A.Abstract The four six–dimensional “little string”theories are all described in the infinite momentum frame (IMF)as matrix theories by non–trivial 1+1dimensional infra–red fixed points.We characterize these fixed points using supergravity.Starting from the matrix theory definition of M5–branes,we derive an associated dual supergravity description of the fixed point theories,arising as the near horizon geometry of certain brane configurations.These supergravity solutions are all smooth,and involve three dimensional Anti–de Sitter space AdS 3.They therefore provide a complete description of the fixed point theories,and hence the IMF little string theories,if the AdS/CFT correspondence holds.9th May 19981.Introduction and Summary1.1.MotivationsRecently[1],it has been shown that all of the ten dimensional superstring theories,de-scribed in the infinite momentum frame(IMF)by the“matrix string”description,have a similar qualitative structure in the region of weak string coupling:•At weak coupling they are all described by1+1dimensional infra–redfixed point theo-ries which are essentially trivial orbifold conformalfield theories.These theories may be described as theflow from an effective1+1dimensionalfield theory:the obvious matrix extension of the relevant Green–Schwarz action,whose prototype was discussed in this framework in ref.[2].•In the same limit,there is an approximate supergravity description,dual(or nearly so)in the sense of ref.[3,4]which is simply the near horizon geometry of the fundamental string solution of a species T–dual to the matrix string in question.•The neighbourhood of the core of the supergravity solution corresponds,via the duality map,to the weak matrix string coupling limit.In the limit,theflow to the trivialfixed point (describing the free matrix string)moves one to the center of the supergravity solution, where the curvature diverges,and the dual description breaks down,as it should.Far away from weak coupling,the matrix descriptions of the strings cease to all resemble one another,and become either0+1dimensional(for the type IIA or E8×E8heterotic systems)or2+1dimensional(for the type IIB or SO(32)type I/heterotic systems).This is of course consistent with the fact that the very strong coupling limits of all of the strings are somewhat different from each other,according to string duality:Thefirst two are dual to eleven dimensional supergravity,while the latter are dual to ten dimensional string theories.In the case of the latter class,the natural description of the theory at intermediate coupling is a2+1dimensional interactingfixed point theory.The theory has a supergravity dual described as eleven dimensional supergravity compactified on AdS4×S7,for the type IIB system,or an orbifold AdS4/Z Z2×S7for the SO(32)system.The isometries of the com-pactification translate into the superconformal symmetries and R–symmetries of the2+1 dimensional conformalfield theory living at the boundary of AdS4.Matrix string theory is a useful alternative way of defining and characterizing string the-ories.In the case of ten dimensions we now have a complete[1]understanding of the overall structure of these theories,and a good understanding of when we can expect a dual supergravity description to help in studying the definingfield theory.There arises the obvious question:What is the analogous story for the more newly discov-ered class of superstring[5,6,2,7,8]theories,the ones which live in six dimensions?These theories have certain properties which make it interesting to begin answering the question.In the light of what was learned for the ten dimensional theories,we can anticipate some of the structure of the matrix–string–via–supergravity description for the six dimensional strings:•The strings all seem to be most naturally defined at intermediate coupling.This is be-lieved to follow from the fact that they are self–dual objects,naturally coupled electrically and magnetically to a three–formfield strength H(3).For this to be true,their coupling is frozen at some value of the coupling of order one.•This means that the strings are always interacting,and therefore we should not expect that the matrix string theory will involve a trivial orbifold conformalfield theory.Instead, there will be some non–trivial interacting theory.This is already known to be true for the (0,2)or“type iia”theory.We will see that it is indeed true for all of the theories.•We should expect further that there should exist a supergravity dual description of the theories.This dual will be complete in the sense that there will be no curvature singularities in the solution,giving us a complete dual theory.For the(0,2)theory,the relevantfixed point is conjectured[3]to be dual to type IIB supergravity compactified on AdS3×S3×T4. We will see that in every case,the AdS3×S3space will arise as the dual,although(of course)the supergravity will be different in each case.We see therefore that the structure of the matrix string definition,or equivalently,the supergravity origin of all of the(IMF)six dimensional string theories is rather simple compared to the ten dimensional theories,precisely because they prefer not to be defined at weak coupling.1.2.Summary of Results•Using the defining matrix theory of longitudinal M5–branes[9],and following the appro-priate limits,we observe that the matrix strings are all defined in terms of1+1dimensional interactingfixed points.(This was already observed for the(0,2)little string[7,10,11].)•The limits which define the matrix string theories also define certain supergravity back-grounds,which can be interpreted as“dual”descriptions in the sense of ref.[3].The dual descriptions are all smooth:⊙The(0,2)theory is given[3]by type IIB supergravity on AdS3×S3×T4.⊙The(1,1)theory comes from type IIA supergravity on AdS3×S3×T4.⊙The(0,1)E8×E8theory is defined by SO(32)heterotic supergravity on AdS3×S3×T4,or alternatively,type IIA supergravity on AdS3×S3×K3.⊙The(0,1)SO(32)theory is defined by E8×E8heterotic supergravity on AdS3×S3×T4.•In all cases therefore,there is the appropriate SO(2,2)bosonic component of the su-perconformal symmetry and SO(4)R–symmetry.The supersymmetry of the relevantsupergravity supplies the appropriate fermionic extension.The R–symmetry has the dual interpretation as the Lorentz group in this light–cone definition of the little strings.•In the two(0,1)cases,the extra SO(32)or E8×E8global symmetries of the little heterotic string theories[8]arise as global symmetries of their definingfixed point theories. These in turn come from the fact that the supergravity compactification will produce a gauge symmetry in AdS3in each case.The AdS/CFT correspondence then demotes this gauge symmetry to a global symmetry of the boundary theory in a similar way to what happens for the Kaluza–Klein gauge symmetries arising from isometries of the S3.2.The case of type iiaWe start with the matrix theory definition of M–theory in the infinite momentum frame (IMF).It is given by[12]the N=16supersymmetric U(N)quantum mechanics arising from N coincident D0–branes’world–volume,in the limitℓs→0and N→∞.The special longi-tudinal direction,x10,(initially compactified on a circle of radius R10),is decompactified in the limit also.The type IIA string theory used to define this theory has parameters:g IIA=R3/210ℓ−3/2p,ℓs=ℓ3/2pR−1/2,10(2.1)whereℓs is the string length andℓp is the eleven dimensional Planck length.Our ultimate goal is to construct the six dimensional(0,2)interacting string theory living on the world volume of a collection of NS–fivebranes of the type IIA theory.Such branes originate from M–theory as M5–branes,transverse to the circle which shrinks to give the type IIA string.Such branes are placed in the matrix theory by adding hypermultiplets to the quantum mechanics,a procedure which is really adding[9]D4–branes to the N D0–brane system in the defining type IIA theory.Let us add M such D4–branes,oriented along the directions x1,...,x4.We need to tune this hypermultiplet theory into its Higgs branch,which is to say we dissolve the D0–branes into the D4–branes,endowing them with N units of D0–brane charge.This system therefore defines M M5–branes oriented along x1,...,x4,x10,with N units of momentum in the x10direction.Following the usual matrix string procedure,we may now imagine that the momentum is actually along the x5direction and shrink that direction to get a definition of the resulting type IIA system.In doing so,we arrive at an economical description of the system by T5–dualizing the defining type IIA system,giving a type IIB string theory configuration consisting of M D5–branes with N D1–branes(or a single D1–brane wound N times onˆx5,the dual direction).The1+1dimensional Yang–Mills coupling on the D1–branes’world–volume is given by: 1/g2YM=ℓ2s/g IIB=ℓ2s R5/R10.As the radius R5shrinks to zero,the ten dimensional type IIB string coupling gets very large.We have a weakly coupled description in terms of theS–dual system of M F5–branes(a shorter term for NS–fivebranes)with N F1–branes (fundamental type IIB strings)inside their world volume.We shall sometimes think of this as one F1–brane with N units of winding in theˆx5direction.After T5–dualizing again,we obtain a type IIA system of M F5–branes with F1–branes (fundamental type IIA strings this time)inside their world–volume with N units of mo-mentum in x5.This chain of dualities is similar to the chain of reasoning which defines the matrix(IMF) ten dimensional type IIA string[2]:There,the defining lagrangian came from a system of D1–branes with N units of winding.This was S–dual to a system of wound type IIB F1–branes.A T–duality on the winding direction gave the type IIA string(F1–brane)with N units of momentum.The Fock space of the IMF matrix string was built up from these winding type IIB strings,and the explicit description at all couplings was given in terms of the D1–brane system,which is a1+1dimensional Yang–Mills theory:a matrix–valued type IIA Green–Schwarz action.At weak coupling,the target space of the theory(moduli space of the1+1dimensional Yang–Mills theory)is simply[13,2]S N(I R8)≡(I R8)N/S N, where S N is the group of permutations of N objects,the D1–branes,and I R8is the space allowed D1–brane positions,the permitted values of the(in general matrix–valued)bosonic fields of the Yang–Mills theory.This is an orbifold theory.As shown in ref.[14,13,2],the twisted sectors of this orbifold describe long type IIA strings which can survive in the large N limit to define strings withfinite momentum in the IMF direction1.The same thing happens here.There is a non–trivial interacting theory living on the type IIA F5–brane’s world–volume even as we take the limit g IIA→0,as argued in ref.[8]. The“little strings”(sometimes called[7,15]“microstrings”)which carry the basic degrees of freedom of the theory are described by the1+1dimensional theory we have defined.It is a1+1dimensional theory derived from the D1–branes’+wrapped2D5–branes’world–volume.The theory has M hypermultiplets in the fundamental of U(N)and it has been tuned to its Higgs branch.In other words,as instantons of the D5–brane’s SU(M)gauge theory,the D1–branes are far from the point–like limit[16]and are instead fat instan-tons,havingfinite scale size.They are delocalized inside the D5–branes,in the directions {x1,...,x4}.Furthermore,we are interested in the zero coupling limit of thefinal type IIA theory and so we should take the strong coupling limit of this configuration.The1+1dimensional Yang–Mills theory is therefore strongly coupled in the limit that we want,and itflows to the infra–red.The resulting infra–redfixed point defines for us the matrix(0,2)“little string theory”.The target space of this theory is[5,7]a hyperK¨a hler deformation of S NM(T4).There has been much work devoted to this theory in the literature[5,17,18,7,10,11,19,20].2.1.The role of Type IIB SupergravityNotice that like the ten dimensional case,we are led to describe the long strings in the theory as winding type IIB F–strings.These long strings arise in the large N limit,which we must take to properly define the original matrix M–theory,and here in order to obtain the light cone type IIA string theory.In this large N limit,we must take seriously the supergravityfields generated by the D1–brane configuration.If we take large M also,we can fully describe the supergravityfields with a metric valid for low curvature everywhere[21]:ds2= 1+g IIBℓ2s N r2 −1/2 (−dt2+dx25)+ 1+g IIBℓ2s M1+g IIBℓ2s Mvr2 1/2NM u2(−dt2+dx25)+du2MThis AdS/CFT correspondence is conjectured to be the full description of the non–trivial conformalfield theory.In this sense,the(0,2)“little string”theory(in the infinite mo-mentum frame)has a supergravity origin.3.The case of type iibThere is a little string theory living on F5–branes of type IIB string theory as well.We should try to characterize it also.Starting again with our matrix definition of M5–branes,we may proceed to descend to type IIB theory,with its F5–branes by compactifying on an additional circle,x4,in addition to the one which we shrunk to get the type IIA theory.We are shrinking M–theory on a torus,and therefore should obtain the type IIB theory[28,29].The extra detail of the M D4–branes in our defining type IIA theory should be interesting.In doing so,we obtain,after T45–dualizing,a type IIA string theory again with M D4–branes located now in{x1,x2,x3,ˆx5}with N D2–branes in{ˆx4,ˆx5}.These D2–branes are delocalized inside the D4–branes,as before.They are not infinite in theˆx4directions, as the D4–branes are pointlike there,and so they end on them.We get the directions {ˆx4,ˆx5}directions both decompactified when we define the resulting type IIB theory at intermediate coupling.To get a weakly coupled type IIB string theory,we would letˆx5 grow faster thanˆx4.In a frame where wefixˆx5large,we see that theˆx4direction shrinks away.We shall do this presently.The effective gauge coupling of the1+1dimensional theory living on the part of the D2–brane is given by:1/g2YM=ℓs/˜g IIA=R4R5/R10.For small R4,R5,both the dual type IIA string theory coupling˜g IIA and the Yang–Mills coupling g YM are large.This means that we should consider our configuration as an M–theory configuration:eleven dimensional supergravity with branes.The M D4–branes become M M5–branes,now stretched along {x1,x2,x3,ˆx5,x10},while the N M2–branes are stretched along{ˆx5,ˆx4}.They end on the M5–branes,and are delocalized inside them.The weakly coupled type IIB string limit,with M F5–branes,is described by takingˆx4→0, giving M F5–branes in type IIA with fundamentalˆx5–wound F–strings inside;T5–duality completes the route to the type IIB system with F–strings possessing momentum in x5. Requiring weakly coupled type IIB therefore focuses the discussion on M5–brane world–volume.The leg of the M2–branes not inside the M5–branes becomes less important to the discussion and the physics of the effective string inside the M5–branes’worldvolume dominates.This1+1dimensional theory therefore describes whatever interacting theory there is on the F5–brane at weak type IIB string coupling.3.1.The role of Eleven Dimensional SupergravityFurthermore,the large N,M limit allows us to discuss the system in terms of the super-gravity solution[30,31]:ds2= 1+ℓ2p N r2 −2/3(−dt2+dx25)+ 1+ℓ2p N r2 1/3(dx21+dx22+dx23+dˆx210) + 1+ℓ2p N r2 −1/3dˆx24+ 1+ℓ2p N r2 1/3(dr2+r2dΩ23).(3.1)(Here,v is a dimensionless measure of the volume of the T4on which the M5–brane is wrapped.r2= 9i=6x2i)In the limit,this solution becomes[32]simply AdS3×S3×T4×S1. It is easy to compute that(after a rescaling)the radius of thefirst two factors is set by the product(MN2)1/3:ds2∼(MN2)1/3ℓ2p u2(−dt2+dˆx25)+du2N 1/3(dx21+dx22+dx23+dˆx210)+Nit is easily established that our metric(3.2)becomes precisely the ten dimensional solution (2.3).We are therefore left with type IIA supergravity3compactified on AdS3×S3×T4.It is natural to conjecture that the AdS/CFT correspondence defines for us a1+1dimensional superconformalfield theory on the boundary with the correct superconformal algebra and R–symmetries as before.Of course,the details of the theory are different,as they should be:This is a different supergravity theory.This1+1dimensional superconformalfield theory defines the(1,1)six dimensional little string theory.Thisfixed point has a dual supergravity solution which is smooth every-where.The type iib system in the infinite momentum frame therefore arises from a simple supergravity description.4.The case of the little E8×E8heterotic string.The next step is obvious.We may place a family of M M5–branes into the E8×E8string theory by introducing M D4–branes into the defining D0–brane system,which additionally contains16D8–branes and2O8–planes4.We orient the eight dimensional objects in {x1,...,x4,x6,...,x9},and the D4–branes in{x1,...,x4},as before.This defines M–theory on an interval(in x5)defined by an M9–plane at each end of it,with M M5–branes located pointwise along it.Everything has momentum in the x10direction.As usual,we can choose to place the momentum in the x5direction,and shrink it.The theory becomes a type IIB system with M D5–branes,with N D1–branes delocalized inside them.The background of16D9–branes has an SO(16)×SO(16)Wilson line.The(0,4)1+1dimensional theory on the world volume of the D1–branes has M hypermul-tiplets from the1–5sector and32fermions from the1–9sector.Without the D5–branes, this theory goes in the strong coupling limit to an IRfixed point which defines the weakly coupled E8×E8heterotic string.In the present case,theflow defines the content of the in-teracting(0,1)six dimensional theory on the world–volume of the F5–brane of the E8×E8 heterotic string theory.This interacting theory has also a global E8×E8symmetry.(See also refs.[39],for related models.)4.1.The role of SO(32)SupergravityIn similar fashion to that which we described for the type iia system,the large N andM limit tells us to examine the supergravityfields around the D1–D5system,but now in type IB string theory.The supergravity solution is precisely the same as it was for the type IIB case.The D5–branes remain small instantons of the D9–brane gauge group and so there is no modification to the supergravity solution by an expression for large instantons.Therefore,in the limit we are led to type IB’s SO(32)supergravity compactified upon AdS3×S3×T4,and in the strong coupling limit(implied by shrinking R5)we replace this with the heterotic SO(32)supergravity with the same compactification(this is valid as the dilaton is constant).Formally,we still need to have winding around theˆx5direction. Near the limit,we can think of it as a large circle,and the spacetime become AdS3in the limit.We can place the required SO(16)×SO(16)Wilson line around this direction,as dictated by the model.The near–AdS3inherits a gauge symmetry SO(16)×SO(16)from ten dimensions.There are states in the adjoint(120,1)+(1,120).We expect that in the presence of the Wilson line,the N→∞approach to AdS3will give masses to states in the(16,16),while states in the(128,1)+(1,128)become massless,performing the expected enhancement to E8×E8 at strong coupling5It is natural to conjecture that there is a non–trivial1+1dimensional superconformalfield theory living at the boundary of the AdS3space.This is the matrix theory of the E8×E8 little heterotic string theory,with infinite momentum frame in the x5direction.In this N→∞limit,the correct long strings should emerge in the usual way.The gauge symmetry anticipated above gives rise to a global E8×E8symmetry of the interacting conformalfield theory living at the boundary,and hence of the six dimensional spacetime little string theory.4.2.The role of Type IIA/Heterotic DualityThere is of course another route to defining the E8×E8little heterotic string.This may also be viewed as a proof of the correctness of the above prescription(particularly the incom-plete argument for the gauge/global symmetry)using type IIA/heterotic duality[40,41,42]. In section4.1we recovered the SO(32)type IB system compactified on AdS3×S3×T4.As a ten dimensional system,we took it to strong coupling.We used ten dimensional SO(32) type IB/heterotic duality to relate this to a weakly coupled heterotic system compactified on the same space.Next,we can consider the system as six dimensional again and replace the T4with a K3while replacing the heterotic theory with the type IIA theory.As shown in ref.[41],the type IIA string recovers the E8×E8symmetry of a dual heterotic string from the intersection lattice of the K3.So simply replacing the T4by a K3in the type IIA supergravity AdS3×S3×T4com-pactification already established in section3.2,we get the E8×E8little heterotic string theory.This way of realizing the string sharpens our earlier discussion of the origin6of the required E8×E8global symmetry,while the duality to the type IB system of the previous subsection points to its correctness.5.The case of the little SO(32)heterotic stringTo define the SO(32)system,we start with the D0–D4–D8–O8theory from the previous section and shrink x4as well as x5.As in the case of type IIB,we arrive at the case of intermediate coupling for the parent theory,if we treat both directions the same way: R4∼R5.Taking the same limits as section3,we arrive at an M–theory configuration involving N M2–branes stretched alongˆx4,ending on M M5–branes which are pointlike along the ˆx4interval defined by the two M9–planes.To get the limit of weak heterotic SO(32)coupling,we would send the size of theˆx4in-terval to zero,and the resulting strongly coupled1+1dimensional theory of the endpoints of the M2–branes inside the M5–branes is the theory we want.In the weakly coupled het-erotic limit,where the size of the interval shrinks away completely,these1+1dimensional endpoints become heterotic F1–strings inside heterotic F5–branes.In taking the limit,we have not tuned things such that the M5–branes stay away from the M9–plane endpoints even in the limit.This would give expectation values to some number of tensor multiplets in the resulting six dimensional theory.Nor have we tuned things such that the F5–branes dissolve into the M9–planes,becoming fat instantons.Instead,we are just at the dividing edge between these branches of the F5–brane moduli space,so that the full E8×E8gauge symmetry is present;the F5–branes are small instantons,and we are able to disappear down the infinite throat tofind the decoupled theory.This is the interacting six dimensional theory we seek.Let us take the large N,M limit,as before.We can then define what the supergravity dual should be.5.1.The role of Eleven Dimensional SupergravityAt intermediate coupling for the SO(32)system,it is easy to see that the large N,M limit of the M2–M5–M9system becomes eleven dimensional supergravity compactified upon AdS3×S3×T4×S1/Z Z2,the boundaries of the orbifoldedˆx4direction are the M9–planes. They are infinitely far apart in the limit R4=0.This should be contrasted with the case of the pure SO(32)system at intermediate coupling encountered in ref.[1].There,the system was defined by AdS4/Z Z2×S7,i.e.,the orbifold acted in the AdS4givingfixed points inside the space7.Here,the system is much simpler,as the orbifold misses the AdS component entirely,due to the presence of the M5–branes.As before,we actually want the weak SO(32)heterotic coupling(g HB=R5/R4)limit,which corresponds to shrinking theˆx4direction.There is no radial dependence of the metric component in this direction,and so the resulting ten dimensional theory is also simple (and non–singular).The relevant supergravity theory is of course now the E8×E8heterotic supergravity[44].5.2.The role of E8×E8SupergravityIn the limit,we have E8×E8supergravity compactified on AdS3×S3×T4.Formally,near the limit we still need to have winding around theˆx5direction,and so We can think of it as an extremely large circle which we will asymptote to the AdS3in the limit.We then place an SO(16)×SO(16)Wilson line in this direction.In the N→∞limit,(and R5→0limit)the long strings emerge in the usual way.We expect also that the correct SO(16)×SO(16)quantum numbers survive to this time recover SO(32)symmetry instead of E8×E8,consistent with the known T5–duality with the Wilson line.Again,we have no direct proof of this mechanism.It is natural to conjecture that there is a non–trivial1+1dimensional superconformalfield theory living at the boundary of the AdS3space.This is the matrix theory of the SO(32) little heterotic string theory,with infinite momentum in the x5direction.6.Closing Remarks and OutlookSo we see that,in the spirit of ref.[1],interacting matrix string theories are captured by smooth dual supergravity compactifications involving AdS.The four little string theories are naturally interacting six dimensional theories obtained by“capturing”ten dimensional strings with F5–branes[8].This is of course why they are four in number:only four of the ten dimensional superstring theories can ensnare such descendents,as the pure type IB system does not contain the requisite F5–branes with which to do the capturing.(In some sense,it also does not contain an honest defining type IB string either:There is no NS–NS B–field.)This is all consistent with what we observe here,because no AdS3geometry arises in the limit of weak type IB coupling.This is because the legs of the M2–brane defining the intermediate coupling situation(see section5.1)are not on the same footing(pun intended) in this case8.One sees that the weak type IB coupling limit(obtained by shrinkingˆx5) would lead back to type IA supergravity(as in ref.[1]),but the AdS3gets spoiled.At best, this leads to a0+1dimensional theory with a singular supergravity limit:presumably a non–interacting theory defined by a quantum mechanics with a trivial orbifold moduli space,following the philosophy of the present paper and ref.[1].The four theories have all been shown to have supergravity defining duals which in-volve AdS3×S3.In each case,the supergravity changes appropriately to give the correct fermionic extension to SO(2,2)tofill out the required supersymmetry algebra for the defining1+1dimensionalfixed point theory on the boundary.In the heterotic cases the supergravity also supplies the required gauge symmetry,although we did not supply a direct argument for how SO(32)and E8×E8get exchanged from an AdS supergravity point of view:Somehow,a more careful examination of the interplay between the Wilson line and the approach to the AdS3×S3geometry(where the circle goes away)should give the required result that the SO(32)system(plus Wilson line)gives E8×E8gauge symmetry in the limit and vice–versa.It is an AdS supergravity analogue of T–duality with Wilson lines.The type IIA/heterotic duality argument presented in section4.2is so far the best direct supergravity argument presented here.At risk of over–emphasizing the point,let us remark upon the simplicity of the overall structure we have uncovered here for all of the string theories,combining the results of ref.[1]and the present paper:•For thefive(IMF)ten dimensional superstring theories near weak coupling,there is a dual description in terms of a solution of the supergravity associated to the T–dual species of string.The solution is merely the near horizon geometry of the fundamental string solution in ten dimensions.•The theories have weak coupling limits where they become free.This is represented by the fact that the fundamental string solution is singular at the core:supergravity must break down there as it cannot describe such a trivial theory.。
World-Sheet Defects, Strings, and Quark Confinement
φ 0 O = Z S (φ 0 )
(1)
where O are operators in the CFT. The important property of AdS is its ‘holographic’ nature, in the sense of uniqueness theorems [26, 3], that specify the bulk behaviour of the classical field (or string) theory of φ in terms of the boundary value φ0 . The conjecture (1) allows for a (non-perturbative) computation of correlation functions of certain supersymmetric conformal field theories, notably some conformal supersymmetric gauge theories in their strong coupling (confining) regime [3]. In view of (1) and the associated holographic nature of AdS , this means that quantum information about the confining physics of a non-Abelian Gauge theory is thus encoded in classical geometries. It should be noticed that in order for the supergravity solutions used in [4, 3] to be trusted one must work in a large Nc limit of a strongly coupled supersymmetric and conformal U (Nc ) gauge theory. The appro2 priate limits are taken in such a way that gY M Nc is fixed but large, as Nc → ∞, 2 with gY M ∼ gS , with gS the string theory coupling. The supersymmetry in the above approach is needed because the entire approach is based on critical-dimension ‘super-string’ theory, in the sense of tensoring the four-dimensional space-time manifold with appropriate compact manifolds, so as to ensure a critical total central charge. It is the purpose of this talk to report on an attempt to extend these developments to a non-critical-dimension string theory, which might open the way for an extension of the above ideas to non-supersymmetric theories. This talk is based on original
The origin of space-time as seen from matrix model simulations：时空的起源,从矩阵模型模拟
Gauge/gravity duality predicts that this should be reproduced by 1d SYM. large-N, low T
microscopic origin of the black hole thermodynamics
quantum description of the states inside the BH
Comparison including corrections
Hanada-Hyakutake-J.N.-Takeuchi, PRL 102 (’09) 191602 [arXiv:0811.3102]
corrections
3. Direct test of gauge-gravity correspondence
Matrix model formulation of superstring/M theories
Banks-Fischler-Shenker-Susskind (2019), Ishibashi-Kawai-Kitazawa-Tsuchiya (2019)
Dynamical origin of space-time Applications to the physics beyond the Standard Model
interactions between D-branes string field theory from SD eqs. for Wilson loops
Fukuma-Kawai-Kitazawa-Tsuchiya (’98) c.f.) Matrix Theory Banks-Fischler-Shenker-Susskind (’96)
duty_report
The topic : The universe is eleven dimension.As we all know:Zero dimension is a point, one dimension is the line, two dimension is the face, three dimension is the space that we know.As for the four-D space, we know that it is determined by the time.We can imagine ourselves now and two years later to two points, connecting these two points is the time, which is the line in four-D space.The four-D space is expanded five-D space. The five-D curved space is six-D space.But if we want to imagine a higher-dimensional space, we must change our mind.Six-D space contains a lot of the time line, the time line guiding the universe toward a different outcome. So a six-D space itself is infinite. This infinite as a point, and then imagine another universe. In which the universe has a different time, gravity, and even the speed of light, for the six-D of the universe, the universe and the infinite number of outcomes constitute a new unlimited. Think this unlimited as a point. Link the two points I have mentioned. This is the line in the seventh-dimensional space.The seventh dimensional bifurcation is the eighth dimension. The eighth dimensional curving is the ninth dimension.All possible universes and all the possible time line as a whole, is ten-dimensional space. Ten-D space contains all the possibilities. Finally, the string theory let us know, since superstring vibrations, the eleventh-dimensional space may exist.Question:What does eight-dimensional space mean?。
N=(4,4) 2D Supersymmetric Gauge Theory and Brane Configuration
a rXiv:h ep-th/9712v28O ct1997IPM-97-2382Oct.1997N =(4,4)2D Supersymmetric Gauge Theory and Brane Configuration Mohsen Alishahiha 1Institute for Studies in Theoretical Physics and Mathematics,P.O.Box 19395-1795,Tehran,Iran Abstract We construct type II A brane configuration of N =(4,4)supersymmetric two dimensional gauge theory with gauge group U (1)and N f hypermultiplets in the fundamental representation.By lifting to M-theory (strong coupling),we can see the origin of the R-symmetry enhancement of the Coulomb branch.One can also find two theories which become equivalent at strong coupling.Brane theory gives us a very useful tool to construct supersymmetric gauge theory in various dimension with various supercharges.Atfirst,Hanany and Witten [1]constructed a particular brane configuration in type II B string theory which describes mirror symmetry in three dimensional supersymmetric gauge theory with 8supercharges.It was shown that there is a duality between the Higgs branch and the Coulomb branch of N=4SYM in three dimension[2].This duality acts as mirror symmetry,exchanging the Higgs and Coulomb branches of the theories. Applying string duality to the configuration of branes in type II B string theory,one can provide an explanation for the mirror symmetry.It has been shown[3]by a particular configuration of type II A string theory that one can obtain supersymmetric U(n)gauge theory in four dimension with four supercharges.Then by making certain deformation in brane configuration, Seiberg’s duality can be realized in brane theory.Introducing orientifold plane in brane configuration help us to generalize this work to other classical Lie gauge groups.[4]N=2four dimensional gauge theory with gauge group SU(n)was also obtained from brane configuration in type II A string theory[5].By lifting from type II A to M-theory,one can obtain the exact solution of N=2D=4SYM theory. More precisely,in M-theory point of view,we have afive brane with worldvolume R3,1×Σ,where the theory on the R3,1space is N=2D=4SYM theory andΣis the Seiberg-Witten curve corresponding to it.This work was generalized to other classical Lie gauge groups in[6].There is another method to studying gauge theory from string theory,the so called Geometric Engineering.By this approach the same class of the above theories can be studied by using wrapped D-branes on Calabi-Yau cycles in type II B and A string theory compactified on the K3-fibered Calabi-Yau3-fold.[7]This approach was generalized in[8]to explain mirror symmetry in N=4three dimensional gauge theory as well as Seiberg-Witten models with product of gauge groups.Recently,Hanany and Hori[9]used brane configuration in type II A string theory to construct N=2supersymmetric gauge theory in two dimension.The aim of this article is to construct brane configuration for N=(4,4)gauge theory in two dimension,through a particular brane in type II A and M-theory.Very recently, this theory was analyzed in[10]and[11].Here we will use three kinds of objects in type II A string theory:NS5-brane with worldvolume(x0,x1,x2,x3,x4,x5)which lives at a point in the(x6,x7,x8,x9)direc-tions.D2-brane with worldvolume(x0,x1,x6)at the point(x2,x3,x4,x5,x7,x8,x9) and also D4-brane with worldvolume(x0,x1,x7,x8,x9)living at the point(x2,x3,x4, x5)and x6.Brane configuration which we will consider is two NS5-branes at r1=(x71,x81,x91), x6=0and r2=(x72,x82,x92),x6=L.A D2-brane suspended between these two NS 5-branes,so it isfinite in the x6direction,and N f D4-branes at m i=(x2i,x3i,x4i,x5i) between two NS5-branes.(See the followingfigure)2,3,4,57,8,96have the Coulomb branch,which means that if we want to have a supersymmetric configuration,the D2-beane must be broken into D2-branes between NS5-branes and D4-branes.If the D4-branes have the same position in x2,x3,x4,x5(equal mass m i=m j),the theory is in the Higgs branch.Note that,for N f=1the D 2-brane must be suspended between D4-brane and two NS5-branes,so thses two D2-brane arefixed from two sides by boundary condition,then the theory does not have the Coulomb and Higgs branches,classically[11].(followingfigure)It was shown[10]that quantum mechanically the theory with N f=1can have Higgs branch although classically it does not have it.The distance between two NS5-branes determines the gauge coupling constant of the two dimensional theory,more precisely1λ(1) whereλis the string coupling constant.The gauge coupling constant can be pro-moted to a background vector superfield,thus it can affect only the metric on the Coulomb branch[11].From our interpretation of mass and Fayet-Iliopoulos termes it is easy to see that[11]:i)The mass terms can be regarded as scalar components of vector superfield,thus can affect only the Coulomb branch.ii)The Fayet-Iliopoulos terms can be promoted to hypermultiplets,thus can affect the metric on the Higgs branch.In the spirit of[1]presence of N f D4-branes induce magnetic charges in(u,v) space.So it can affect the metric on the Coulomb branch.2Minimizing the totalfive brane worldvolume,wefind four dimensional Laplace equation which has solution of the form A+Bg2+N fg2+1|X−m2|2+···+12Note that in two dimension,moduli space of vacua is not well defined.But in the spirit of the Born-Oppenheimer approximation,on can refer to it as a target space on non-linear sigma model[11]which is one-loop correction to the metric in the Coulomb branch[11].Therefore, the generalized Kahler potential is[13]K=1u¯u dη4N f sin2θ2v=e i(χ+φ)/2sinθReferences[1]A.Hanany,E.Witten,Nucl.Phys.B492(1997)152.[2]K.Intriligator,N.Seiberg,Phys.Lett.B387(1996)513.[3]S.Elitzur,A.Giveon,D.Kutasov,hep-th/9702014.[4]N.Evans,C.V.Johnson,A.D.Shapere,hep-th/9703210.S.Elitzur, A.Giveon, D.Kutasov, E.Rabinovici, A.Schwimmer,hep-th/9704104.R.Tatar,hep-th/9704198.J.H.Brodie,A.Hanany,hep-th/9704043.A.Hanany,A.Zaffaroni,hep-th/9706047.A.Brandhuber,J.Sonnenschein,S.Theisen,S.Yankielowicz,hep-th/9704044E.Witten,hep-th/9706109A.Brandhuber,N.Itzhaki,V.Kaplunovsky,J.Sonnenschein,S.Yankielowicz,hep-th/97060127K.Hori,H.Ooguri,Y.Oz,hep-th/9706082.[5]E.Witten,hep-th/9703166.[6]ndsteiner,E.Lopez,D.A.Lowe,hep-th/9705199.A.Brandhuber,J.Sonnenschein,S.Theisen,S.Yankielowicz,hep-th/9705232.[7]S.Kachru,C.Vafa,Nucl.Phys.B450(1995)69.A.Klemm,W.Lerche,P.Mayr,C.Vafa,N.Warner,Nucl.Phys.B477(1997)746.H.Ooguri,C.Vafa,Nucl.Phys.B463(1996)55.S.Katz,A.Klemm,C.Vafa,hep-th/9609239.[8]K.Hori,H.Ooguri,C.Vafa,hep-th/9705220.S.Katz,P.Mayr,C.Vafa,hep-th/9706110.H.Ooguri,C.Vafa,hep-th/9702180.C.Ahn,K.Oh,hep-th/9704061.C.Ahn,hep-th/9705004.C.Ahn,R.Tatar,hep-th/9705106.[9]H.Hanany,K.Hori,hep-th/9707192.[10]E.Witten,hep-th/9707093.[11]D.E.Diaconescu,N Seiberg,hep-th/9707158.[12]E.Witten,hep-th/9507121.(World Scientific,1996).[13]M.Rocek,K.Schoutens,A.Sevrin,Phys.Lett.B265(1991)303.M.Rocek,C.Ahn,K.Schoutens,A.Sevrin,hep-th/9110035.D.E.Diaconescu,N Seiberg,hep-th/9707158.[14]J.H.Brodie,hep-th/9709228.。
An Alternative String Theory in Twistor Space for N=4 Super-Yang-Mills
a r X i v :h e p -t h /0402045v 1 5 F eb 2004An Alternative String Theory in Twistor Spacefor N=4Super-Yang-MillsNathan Berkovits 1Instituto de F´ısica Te´o rica,Universidade Estadual Paulista Rua Pamplona 145,01405-900,S˜a o Paulo,SP,BrasilIn this letter,an alternative string theory in twistor space is proposed for describ-ing perturbative N=4super-Yang-Mills theory.Like the recent proposal of Witten,this string theory uses twistor worldsheet variables and has manifest spacetime superconformal invariance.However,in this proposal,tree-level super-Yang-Mills amplitudes come from open string tree amplitudes as opposed to coming from D-instanton contributions.February 2004In a recent paper[1],Witten has shown that the simple form of maximal helicity violating amplitudes of Yang-Mills theory has a natural generalization to the non-maximal helicity violating amplitudes.He also constructed a topological B-model from twistor worldsheet variables and argued that D-instanton contributions in this model reproduce the perturbative super-Yang-Mills amplitudes.For details on this model and the twistor approach to super-Yang-Mills,see the review and references in[1].The formula for D-instanton contributions of degree d to n-gluon tree-level amplitudes is[1][2]B(λr,¯λr)= d2d+2a d2d+2b d4d+4γn r=1 dσr(vol(GL(2))−1(1) n−1r=1(σr−σr+1)−1(σn−σ1)−1n r=1δ(λ2rλ1(σr))exp(i¯λ˙αrλ1rµ˙α(σr)λ1(σ1))φ2(ψA(σ2)λ1(σn))]where Pα˙αr=λαr¯λ˙αr is the momentum of the r th state,λα(σ)=dk=0aαkσk,µ˙α(σ)=d k=0b˙αkσk,ψA(σ)=d k=0γA kσk,φr(ψA)is the N=4superfield whose lowest component is the positive helicity gluon and whose top component is the negative helicity gluon,and the(vol(GL(2))−1factor can be used to remove one of the a integrals and three of theσintegrals.For maximal helicity violating amplitudes(i.e.n−2positive helicity gluons and2 negative helicity gluons),the above formula when d=1has been shown to give the correct n-point amplitude.For non-maximal helicity violating amplitudes,it has been suggested that this formula may also give the correct n-point amplitude where one has n−d−1 positive helicity gluons and d+1negative helicity gluons.Although there is a possibility that the formula of(1)needs to be modified for non-maximal amplitudes by contributions from instantons of lower degree,it has been recently verified that no such modifications are necessary when d=2and n=5[2].It will be assumed below that the formula of(1) correctly reproduces the super-Yang-Mills tree amplitudes for any d and n.In this letter,a new string theory in twistor space is proposed which reproduces the formula of(1)using ordinary open string tree amplitudes as opposed to D-instanton contributions.This string theory shares many aspects in common with the orginal ideaof Nair in[3].The worldsheet matter variables in this string theory consist of a left and right-moving set of super-twistor variables,Z I L=(λαL,µ˙αL,ψA L),Z I R=(λαR,µ˙αR,ψA R)(2)forα,˙α=1to2and A=1to4,a left and right-moving set of conjugate super-twistor variables,Y LI=(¯µLα,¯λL˙α,¯ψLA),Y RI=(¯µRα,¯λR˙α,¯ψRA),(3) and a left and right-moving current algebra,j C L,j C R(4)where C is Lie-algebra valued and j C L and j C R satisfy the usual OPE’s of a current algebra, i.e.j C L(y)j D L(z)→f CDE j EL(z)(y−z)2,j C R(¯y)j D R(¯z)→f CDE j ER(¯z)(¯y−¯z)2.(5)The current algebra can be constructed from free fermions,a Wess-Zumino-Witten model, or any other model.The worldsheet action isS= d2z(Y LI∇R Z I L+Y RI∇L Z I R)+S G(6) where S G is the worldsheet action for the current algebra and(∇R,∇L)contains a world-sheet GL(1)connection for which Z I L and Z I R have charge+1,and Y LI and Y RI have charge−1.Quantizing this worldsheet action gives rise to left and right-moving Virasoro ghosts, (b L,c L)and(b R,c R),as well as left and right-moving GL(1)ghosts,(u L,v L)and(u R,v R). The untwisted left-moving stress tensor isT0=Y LI∂L Z I L+T G+b L∂L c L+∂L(b L c L)+u L∂L v L(7) where T G is the left-moving stress tensor for the current algebra,and the left-moving GL(1) current isJ=Y LI Z I L.(8)To have vanishing conformal anomaly,the current algebra must be chosen such that the central charge contribution from T G is28.Note that there is no GL(1)anomaly because of cancellation between bosons and fermions in J.The open string theory is defined using the conditionsZ I L=Z I R,Y LI=Y RI,j C L=j C R,c L=c R,b L=b R,v L=v R,u L=u R(9)on the open string boundary.Unlike a usual open string theory where Lie algebra indices come from Chan-Paton factors,the Lie algebra indices in this open string theory come from a current algebra.The physical integrated and unintegrated open string vertex operator for the super-Yang-Mills states isV= dz j C(z)ΦC(Z(z)),U=c(z)j C(z)ΦC(Z(z)).(10) The superfieldsΦC(Z)are similar to those defined in[1],namely for a super-Yang-Millsstate with momentum Pα˙αr=λαr¯λ˙αr,ΦC(Z(z r))=δ(λ2rλ1(z r))exp(i¯λ˙αrλ1rµ˙α(z r)λ1(z r))(11)whereφC(ψA)is the same N=4superfield as in(1).Note thatΦC(Z)is GL(1)-neutral and has zero conformal weight.Tree-level open string scattering amplitudes are computed in the usual manner from the disk correlation functionA= U1(z1)U2(z2)U3(z3) dz4V4(z4)... dz n V n(z n) (12) where different twistings of the stress tensor are used to compute different helicity violating amplitudes.For amplitudes involving(n−d−1)positive helicity gluons and d+1negative helicity gluons,the twisted stress tensor is defined asT d=T0+d2and Y I has conformal weight d+2bosonic and4d+4fermionic zero modes of Z I,except for the one bosonic zero mode which can be removed using the worldsheet GL(1)gauge invariance.Performing the correlation function for the current algebra gives the contribution2T r[φ1...φn]n−1r=1(z r−z r+1)−1(z n−z1)−1,(15)and the(b,c)correlation function gives the factor(z1−z2)(z2−z3)(z3−z1).So one obtains the formulaA= d2d+2a d2d+2b d4d+4γ dz1... dz n(V ol(GL(2)))−1(16) n−1r=1(z r−z r+1)−1(z n−z1)−1n r=1δ(λ2rλ1(z r))exp(i¯λ˙αrλ1rµ˙α(z r)λ1(z1))φ2(ψA(z2)λ1(z n))]whereλα(z)=dk=0aαk z k,µ˙α(z)=d k=0b˙αk z k,ψA(z)=d k=0γA k z k,(aαk,b˙αk,γA k)are the zero modes of Z I on a disk,and the SL(2)part of GL(2)can be used tofix three of the z r integrals and reproduce the(b,c)correlation function.This formula clearly agrees with the formula of(1)for the D-instanton amplitude where theσvariable from the D1-string worldvolume has been replaced with the z variable from the open string boundary.Acknowledgements:I would like to thank Cumrun Vafa,Peter Svrcek and espe-cially Edward Witten for useful discussions and the Institute for Advanced Study for their hospitality andfinancial support.I would also like to thank CNPq grant300256/94-9, Pronex66.2002/1998-9,and Fapesp grant99/12763-0for partialfinancial support.References[1] E.Witten,Perturbative Gauge Theory as a String Theory in Twistor Space,hep-th/0312171.[2]R.Roiban,M.Spradlin and A.Volovich,A Googly Amplitude from the B-Model inTwistor Space,hep-th/0402016.[3]V.P.Nair,A Current Algebra for some Gauge Theory Amplitudes,Phys.Lett.B214(1988)215.。
Superspace Type II 4D Supergravity from Type II Superstring
presented here are a generalization of the methods developed for the heterotic superstring in [7]. The core of the hybrid formalism lies in the fact that a critical N=1 string can be formulated as a critical N=2 string, without changing the physical content [9]. This is achieved by twisting the ghost sector of the critical N=1 string . After performing this embedding for the critical RNS superstring, a field redefinition allows the resulting N=2 string to be made manifestly spacetime supersymmetric for compactifications to four dimensions. In this case, the critical c = 6 matter sector splits into a c = −3 four-dimensional part and a c = 9 compactification-dependent part. The action for this superstring can be written as : S = S4D + α′ SF T + Sc , where the first term describes the classical four dimensional part, the second one is the Fradkin-Tseytlin term which has the dilaton coupling and Sc is the action for the compactification-dependent c = 9 theory. This work does not concern about the fields that depend on compactification, so it is enough to concentrate just in the N = (2, 2) c = −3 sector. In a flat 4d background, the type II superstring is described, in the N = (2, 2) superconformal gauge, by the following action: 1 α ˙ α d2 z ∂xm ∂xm + pα ∂θα + pα ˙ ∂θ + +pα ∂ θ 2 α′ α′ α ˙ ∂ρ (∂ρ + az ) + ∂ ρ(∂ ρ + az ) + Sc + α′ SF T . + pα ˙ ∂θ − 2 2 1 α′
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For type-II the central charge of the supersymmetric right-movers is also
cR = 15. However, for the heterotic string the bosonic part SO (3, 2)−k /SO (3, 1)−k gives cR (bose) = 6k 10k − = 15 k−3 k−2 (3)
USC-91/HEP-B6 November 1991
A SUPERSTRING THEORY IN FOUR CURVED SPACE-TIME DIMENSIONS
∗
arXiv:hep-th/9111040v1 20 Nov 1991
I. BARS and K. SFETSOS Physics Department University of Southern California Los Angeles, CA 90089-0484, USA
3
Perhaps the most interesting one is
E7 × SU (3) × SU (2) × U (1) (all at level k = 1) since it contains just the gauge group of The second piece in the action S1 gauges [9] the Lorentz subgroup H = SO (3, 1) which is embedded in SO (3, 2)L × SO (3, 2)R with a deformation. As explained in [4] the action of the gauge group could be deformed on the left or the right of the group element g . If the matrix representation of the gauged Lorentz algebra on the left is ta and the one on the
S0 (g ) =
k 8π
M
d2 σ T r (g −1 ∂+ g g −1 ∂− g ) −
M
k 24π
T r (g −1 dg g −1 dg g −1 dg )
B
k S1 (g, A) = − 4π S2 (ψ+ , A− ) = −
˜+ g −1 ∂− g + A− g A ˜+ g −1 − A− A+ ) d2 σ T r (A− ∂+ gg −1 − A
this piece has SO (3, 2)L × SO (3, 2)R symmetry. Since SO (3, 2) has a non-Abelian compact subgroup SO (3) the quantum path integral could be defined uniquely only for k = integer 2
models that generalize these were also expressed as Kazama-Suzuki cosets of the form model was interpreted as a string propagating on the manifold of a two dimensional black hole [2]. Similarly, in three dimensions the bosonic model describes the ADS string propagating in a singular and more complicated manifold which is a new solution to Einstein’s equations with matter [3] [4] [5] . These manifolds have the interesting property of duality which signals a shortest distance in string theory. The string geodesics, which correspond to the classical solutions of the gauged WZW model, have been obtained in [4] for any dimension, including d = 4. In this paper we will investigate the most interesting case. Namely, the supersymmetric ADS string in four dimensions and k = 5 which in a certain sense is unique. We construct heterotic and type-II string theories based on this coset. The action for our model, in the conformal gauge, has four parts S = S0 + S1 + S2 + S3 with SO (d − 1, 2)−k × SO (d − 1, 1)1 /SO (d − 1, 1)−k+1 for d = 2, 3, · · · , 15 . The d = 2 bosonic
by the total Virasoro central charge for the supersymmetric left movers [1] cL = 3kd = 15 2(k − d + 1) for d = 4, k = 5. (2)
cL is cancelled by the super ghost system of S4 . It is seen that d = 4 is unique in the sense that, for d < 10, it is the only case with integer k solution. The other acceptable flat string. integer k solution occurs for d = 10 (k = ∞) which is the original Neveu-Schwarz-Ramond
In the above, S0 is the global WZW model [6] with g (σ +, σ − ) ∈ SO (3, 2). By itself
(this was not a restriction for d = 2, 3).
1
Indeed, we take k = 5 which is the value required
µ follows if χa − is removed and replaced by ψ− that appears with a gauge covariant kinetic µ . Then S3 , S5 are replaced by S3 (ψ− , A+ ) and S5 (bR , cR , βR , γR ). term just like ψ+
require a cR (χ) = 11 contribution from the free fermions χa − . Therefore the action S3
for k = 5 (already fixed in the action). Since the ghosts in S5 (bR , cR ) contribute −26 we
contains 22 free fermions. This action could be viewed as giving rise to SO (22)1 current algebra theory for right movers. There are many other ways of obtaining cR = 11 as exact conformal theories based on current algebras. the Standard Model and a “hidden” E7 .
∗
Research supported in part by the U.S. Department of Energy, under Grant No. DE-FG03-
84ER-40168
1
Sometime ago exact conformal theories based on the anti-de Sitter (ADS) cosets curved space-time in d = 2, 3, · · · , 26 dimensions [1]. SO (d − 1, 2)−k /SO (d − 1, 1)−k were introduced as models for strings propagating in The N = 1 superconformal
µ d2 σ ψ+ (iD− ψ+ )ν ηµν ,
k 4π
S3 (χ− ) =
M
k 4π
22
d2 σ
M a=1
a χa − i∂+ χ−
(1) In addition, there are ghost actions S4 (bL , cL , βL , γL ) for left movers and S5 (bR , cR ) for right movers that are added due to the superconformal or conformal gauge fixing respectively. This action has (1, 0) superconformal symmetry (see below) and is appropriate for the heterotic string. The type-II string requires (1, 1) superconformal symmetry. Its action